chapter 2.4 again iteration theory -...

Chapter 2.4Again iteration theory

As already mentioned, this book grew from an attempt to generalize the Wolff-Denjoytheorem to several variables, and this chapter is a report of the up-to-date situation in thefield (at least as far as we know). We shall start with iteration theory on taut manifolds(of course); using the foundations laid down in chapters 2.1 and 2.3, it will be very easyto give a necessary and sufficient condition for the convergence of the sequence of iteratesof a holomorphic map, already anticipated by Theorem 2.2.32. A little more work will benecessary to prove that if X is a taut manifold and f ∈ Hol(X,X) is such that {fk} is notcompactly divergent, then {fk} is relatively compact in Hol(X,X); using this, we shall beable to characterize the set of limit points of {fk}.

The complete generalization of the Wolff-Denjoy theorem, that is the description ofthe behavior of a compactly divergent sequence of iterates in a domain, is not so easy.Simple examples show that the naıve statement does not hold either in generic stronglypseudoconvex domains or in weakly convex domains, and it is then natural to wonderwhether it is true at all, or whether it was a pure one-dimensional phenomenon. So, themost important result of this chapter is Theorem 2.4.23: if D ⊂⊂ Cn is a strongly convexdomain and f ∈ Hol(D,D) has no fixed points, then the sequence of iterates of f converges,uniformly on compact sets, to a constant map x ∈ @D.

The proof of this theorem makes use of two distinct tools. The first one are thehorospheres. We already saw in the first part of this book how to use Proposition 1.2.2 todefine horocycles in multiply connected domains; now we shall do something very similarstarting from Proposition 2.2.20, and replacing the Bergmann distance by the Kobayashidistance. Unfortunately, in general the limit (2.2.26) does not exist; so we are forcedto introduce two families of horospheres, using lim inf and lim sup instead of the simple-minded limit. We shall be able to prove new versions of Julia’s and Wolff’s lemmas, andwe shall use them respectively in the study of angular derivatives (in the last chapter) anditeration theory (guess where). Furthermore, the boundary behavior of the horosphereswill be of the greatest importance: in fact, it will turn out that the lack of a Wolff-Denjoytheorem for weakly convex domains is due to the boundary shape of the horospheres.

The second tool is Theorem 2.4.20: if D ⊂⊂ Cn is a convex domain and f is aholomorphic map of D into itself, then {fk} is compactly divergent iff f has no fixedpoints. The proof of this theorem is a mixture of holomorphic (via the Kobayashi distance)and topological (via Brouwer’s theorem) arguments, and indeed it seems that the lack of aWolff-Denjoy theorem for generic (i.e., not homeomorphic to a ball) strongly pseudoconvexdomains is due to topological reasons.

Using these tools we shall be able to study in detail iteration theory of holomorphicmaps in convex domains. In particular, we shall prove the aforementioned generalizationof the Wolff-Denjoy theorem in strongly convex domains, which is one of the highest peaksof this book.

210 2.4 Again iteration theory

2.4.1 Taut manifolds

We start this chapter by discussing, of course, iteration theory of holomorphic maps ontaut manifolds. Let X be a taut manifold; we shall give a complete characterization ofmaps f ∈ Hol(X,X) such that the sequence of iterates {fk} converges in Hol(X,X);furthermore, we shall describe the set of limit points of {fk} in Hol(X,X) for a genericf ∈ Hol(X,X), and we shall also discuss the case of compact hyperbolic manifolds.

Clearly, the main tool here is Theorem 2.1.29, together with the related concepts oflimit retraction and limit manifold. Using them, we can immediately give the characteri-zation of maps with converging sequence of iterates:

Theorem 2.4.1: Let X be a taut manifold, and take f ∈ Hol(X,X). Then the se-quence of iterates {fk} converges in Hol(X,X) iff f has a fixed point z0 ∈ X suchthat sp(dfz0) ⊂ ∆ ∪ {1}.

Proof: Assume first that the sequence {fk} converges, necessarily (by Theorem 2.1.29) tothe limit retraction ρ:X → M . Since

f ◦ ρ = limk→1

f ◦ fk = limk→1

fk+1 = ρ,

f restricted to M is the identity. Take z0 ∈ M ; since dρz0 = limk→1

(dfz0)k, it follows that if

∏ is an eigenvalue of dfz0 then the sequence {∏k} of powers of ∏ converges to an elementof sp(dρz0) ⊂ {0, 1}. Thus ∏ ∈ ∆ ∪ {1}, and the first part of the assertion is proved.

Conversely, assume f has a fixed point z0 ∈ X such that sp(dfz0) ⊂ ∆ ∪ {1}; inparticular, {fk} is relatively compact in Hol(X,X). Let ρ:X → M be the limit retractionof f ; note that M is taut, by Lemma 2.1.15. Arguing as in the proof of Theorem 2.2.32, letTz0X = LN⊕LU be the dfz0-invariant splitting of Tz0X constructed in Theorem 2.1.21.(iv);note that, by Corollary 2.1.30, LU = Tz0M . Since, by hypothesis, dfz0 |LU = id, it followsthat (dfz0)k → dρz0 as k → +1. In particular, z0 is fixed by every limit point h of {fk},and dhz0 = dρz0 . Thus, by Theorem 2.1.21.(iii), h|M = idM and, by Theorem 2.1.29,h = ρ. In other words, ρ is the unique limit point of {fk} and, being {fk} relativelycompact in Hol(X,X), fk → ρ, q.e.d.

So, again, maps with a fixed point arise on the scene. A natural question is: underwhat conditions does the sequence {fk} converge to a point z0 ∈ X? The answer liesin the following definition: an attractive fixed point for a map f ∈ Hol(X,X) is a fixedpoint z0 ∈ X of f such that sp(dfz0) ⊂ ∆. Then

Corollary 2.4.2: Let X be a taut manifold, and f ∈ Hol(X,X). Then {fk} converges toa point z0 ∈ X iff z0 is an attractive fixed point for f .

Proof: If z0 is an attractive fixed point for f , then the limit multiplicity of f is 0 (byCorollary 2.1.30), and so {fk} converges to z0. Conversely, if {fk} converges to z0 ∈ X,then z0 is the limit manifold of f , and the assertion again follows from Corollary 2.1.30,q.e.d.

2.4.1 Taut manifolds 211

Now we would like to describe the limit points of the sequence of iterates of ageneric f ∈ Hol(X,X). If {fk} is compactly divergent, for the moment there is nothing tosay. If {fk} is not compactly divergent, we know that every limit point in Hol(X,X) is ofthe form ∞ ◦ ρ, where ρ:X → M is the limit retraction of f , and ∞ ∈ Aut(M). However, apriori there may be other limit points; a priori, the sequence {fk}, which is not compactlydivergent, could contain compactly divergent subsequences. In other words, it is conceiv-able that {fk} be not relatively compact in Hol(X,X). Fortunately, this is not the case,as shown in the following

Theorem 2.4.3: Let X be a taut manifold, and take f ∈ Hol(X,X) such that {fk} isnot compactly divergent. Then {fk} is relatively compact in Hol(X,X).

Proof: Let M be the limit manifold of f . To prove that {fk} is relatively compactin Hol(X,X), it clearly suffices to show that {(f |M )k} has no compactly divergent subse-quences; so, by Corollary 2.1.31, we can directly assume X = M , that is we can assumef is a pseudoperiodic automorphism of X.

Take z0 ∈ X; it suffices to show that A = {fk(z0) | k ∈ N} is contained in a compactsubset of X. Choose η0 > 0 such that Bk(z0, η0) is compact, where we recall that Bk(z0, η0)denotes the Kobayashi ball of center z0 and radius η0. Since f ∈ Aut(X), it follows thatBk

°fh(z0), η0

¢⊂⊂ X for every h ∈ N. Now

Bk(z0, η0) ⊂ Bk

°Bk(z0, 7η0/8), η0/4

¢,

by Lemma 2.3.15; hence there are w1, . . . , wr ∈ Bk(z0, 7η0/8) such that

Bk(z0, η0) ∩A ⊂r[

j=1

Bk(wj , η0/4) ∩A,

and we can assume Bk(wj , η0/4)∩A 6= /∞ for every j = 1, . . . , r. For each j = 1, . . . , r takekj ∈ N such that fkj (z0) ∈ Bk(wj , η0/4); then

Bk(z0, η0) ∩A ⊂r[

j=1

£Bk

°fkj (z0), η0/2

¢∩A

§. (2.4.1)

Now, since f is pseudoperiodic, the set©k ∈ N

ØØ kX

°fk(z0), z0

¢< η0/2

™is infinite;

therefore we can find k0 ∈ N such that

k0 ≥ max{1, k1, . . . , kr}, (2.4.2)

kX

°fk0(z0), z0

¢< η0/2. (2.4.3)

Put

T =k0[

h=1

Bk

°fh(z0), η0

¢;


since every Bk

°fh(z0), η0

¢is relatively compact in X, it suffices to show that A ⊂ T .

Take h ∈ N. If h ≤ k0, then clearly fh(z0) ∈ T ; so assume h > k0. Choose h0 ≥ hsuch that kX

°fh0(z0), z0

¢< η0/2; hence, by (2.4.3), kX

°fh0(z0), fk0(z0)

¢< η0 and

∀ 0 ≤ j ≤ k0 kX

°fh0−j(z0), fk0−j(z0)

¢= kX

°fh0(z0), fk0(z0)

¢< η0.

In particular,

∀h0 − k0 ≤ j ≤ h0 f j(z0) ∈ T, (2.4.4)

and fh0−k0(z0) ∈ Bk(z0, η0) ∩A. By (2.4.1) there is 1 ≤ l ≤ r such that

kX

°fkl(z0), fh0−k0(z0)

¢< η0/2,

and so

∀ 0 ≤ j ≤ min{kl, h0 − k0} kX

°fh0−k0−j(z0), fkl−j(z0)

¢< η0/2. (2.4.5)

In particular, if kl ≥ h0−k0 then, by (2.4.2), (2.4.4) and (2.4.5), it follows that f j(z0) ∈ Tfor all 0 ≤ j ≤ h0.

If kl < h0−k0, set h1 = h0−k0−kl; then, by (2.4.2), 0 < h1 < h0. Moreover, (2.4.2),(2.4.4) and (2.4.5) imply that f j(z0) ∈ T for h1 ≤ j ≤ h0 and that kX

°fh1(z0), z0

¢< η0/2.

Then we can repeat the argument replacing h0 by h1, and in a finite number of steps weget f j(z0) ∈ T for every 0 ≤ j ≤ h0. In particular, fh(z0) ∈ T ; being h arbitrary, it followsthat A ⊂ T , and the proof is complete, q.e.d.

Then let X be a taut manifold, take f ∈ Hol(X,X) such that the sequence {fk} isnot compactly divergent, and denote by Γ(f) the set of limit points of {fk} in Hol(X,X)(note that the closure of {fk} in Hol(X,X) is Γ(f) ∪ {fk}). By Theorem 2.4.3, Γ(f) isa compact topological semigroup and, by Theorem 2.1.29, it is isomorphic to a compacttopological semigroup of the automorphism group of the limit manifold of f . But evenmore is true:

Corollary 2.4.4: Let X be a taut manifold, take f ∈ Hol(X,X) such that the se-quence {fk} is not compactly divergent, and let ρ:X → M be its limit retraction. ThenΓ(f) is isomorphic to a compact abelian subgroup of Aut(M), which is the closed subgroupgenerated by ϕ = f |M ∈ Aut(M).

Proof: By Theorems 2.1.29, 2.4.3 and Corollary 2.1.31, it remains to show that for eachlimit point h = ∞ ◦ ρ ∈ Γ(f) we have ∞−1 ◦ ρ ∈ Γ(f). But indeed fix a subsequence {fk∫}converging to ρ, and a subsequence {fm∫} converging to h. As usual, we can assume thatk∫ −m∫ → +1 and fk∫−m∫ → h1 = ∞1 ◦ ρ as ∫ → +1; then h ◦ h1 = ρ = h1 ◦ h, thatis ∞1 = ∞−1, q.e.d.

2.4.1 Taut manifolds 213

In general, this is the best we can do. Indeed, let

D =©(z, w) ∈ C2

ØØ |z|2 + |w|2 + |w|−2 < 3™;

D is a strongly pseudoconvex domain, thus taut. Define f ∈ Hol(D,D) by

f(z, w) = (z/2, eiθwε), (2.4.6)

where ε = ±1 and θ ∈ R. Then the limit manifold of f is clearly the annulus

M =©(0, w) ∈ C2

ØØ |w|2 + |w|−2 < 3™,

the limit retraction is ρ(z, w) = (0, w) and choosing opportunely ε and θ we can obtainas Γ(f) any compact abelian subgroup of Aut(M).

However, if f has a fixed point we can mime the proof of Proposition 2.2.33 to getsome more information:

Corollary 2.4.5: Let X be a taut manifold, and take f ∈ Hol(X,X) with a fixedpoint z0 ∈ X. Let ∏1, . . . ,∏r ∈ @∆ be the eigenvalues of modulus 1 of dfz0 , listed ac-cording to their multiplicity; in particular, r is the limit multiplicity of f . Then Γ(f) isisomorphic to a compact abelian group whose connected component at the identity is areal torus group of dimension at most r. To be precise, Γ(f) is isomorphic to the closedsubgroup of Tr generated by (∏1, . . . ,∏r) ∈ Tr.

Proof: By Theorem 2.1.29 and Corollary 2.1.22, every element of Γ(f) is determined bythe restriction to Tz0M of its differential at z0, where M is the limit manifold of f . Theassertion then follows from Theorem 2.1.21.(iv), Corollaries 2.1.30 and 2.4.4, and from thefact that every closed connected subgroup of Tr is a torus group of smaller dimension,q.e.d.

Note that this corollary allows us to describe Γ(f) just looking at the spectrum ofthe differential of f at a fixed point; in particular, contrarily to Corollary 2.4.4, it is notnecessary to know beforehand the limit manifold of f .

If X is compact, then no sequence in Hol(X,X) can be compactly divergent, of course.A consequence of this trivial observation is that on compact taut manifolds the iterationtheory degenerates, as we shall now describe.

We need some information about the structure of Aut(X). The first fact is a verygeneral theorem:

Theorem 2.4.6: Let X be a compact complex manifold. Then Aut(X) is a complex Liegroup.

A proof is in Montgomery and Zippin [1955], for instance.If X is a compact hyperbolic manifold, Aut(X) is a very particular kind of complex

Lie group:


Corollary 2.4.7: Let X be a compact hyperbolic manifold. Then Aut(X) is finite.

Proof: First of all we prove that Aut(X) is discrete. If not, let ∞:C → Aut(X) be aone-parameter subgroup; then for every z0 ∈ X the map ≥ 7→ ∞(≥)z0 is a holomorphic mapof C into X, and hence constant, for kC ≡ 0 and X is hyperbolic. Since this holds forall z0 ∈ X, the only one-parameter subgroup of Aut(X) is the trivial one, and Aut(X) isdiscrete.

To end the proof it is enough to show that Aut(X) is compact. But X is completehyperbolic, and thus taut; therefore (Proposition 2.1.24) Aut(X) is closed in C0(X,X).Since it is clearly equicontinuous with respect to kX , by the Ascoli-Arzela theorem Aut(X)is compact, q.e.d.

Another general fact we shall need is Remmert’s theorem:

Theorem 2.4.8: Let f :X → Y be a proper holomorphic map between two complexanalytic spaces X and Y . Then f(X) is a complex analytic subspace of Y .

A proof can be found in Narasimhan [1966]; note that if X is compact then thistheorem can be applied to any f ∈ Hol(X,Y ), for in this case any element of Hol(X,Y ) isautomatically proper.

Using these powerful tools we can show why iteration theory on compact hyperbolicmanifolds is void, generalizing Corollary 1.3.13:

Theorem 2.4.9: Let X be a compact hyperbolic manifold, and f ∈ Hol(X,X). Thenthere exists m ∈ N∗ such that fm is a holomorphic retraction. In particular, the sequenceof iterates of f converges iff f itself is a holomorphic retraction.

Proof: Let ρ:X → M be the limit retraction of f . By Corollary 2.1.31, f |M ∈ Aut(M);in particular, fk(X) ⊃ M for every k ∈ N.

Now, X ⊃ f(X) ⊃ f2(X) ⊃ · · · is a descending chain of compact hyperbolic ana-lytic spaces; by Theorem 2.3.41 (cf. the notes to chapter 2.3), at every stage f is eithera biholomorphism or everywhere degenerate. Therefore there is a k0 ∈ N such thatfk+1(X) = fk(X) = M for all k ≥ k0.

By Corollary 2.4.7, Aut(M) is finite; so there is m ≥ k0 such that (f |M )m = idM .But then for any z ∈ X we have

f2m(z) = fm°fm(z)

¢= fm(z),

since fm(z) ∈ M , and we are done, q.e.d.

The next step in iteration theory is the study of the behavior of compactly divergentsequences of iterates in domains of Cn. The candid hope in an immediate generalizationof the Wolff-Denjoy theorem is undeceived at once: the map f defined in (2.4.6) has nofixed points and yet {fk} is not converging. An even worse example is the following: letf ∈ Hol(∆2,∆2) be given by

f(z, w) =µ

1 + z

3− z, eiθw

∂,

2.4.2 Horospheres 215

for some θ ∈ R with eiθ 6= 1. Then f is fixed point free, the sequence {fk} is evencompactly divergent, and yet {fk} does not converge. As we shall see in the next twosections, this behavior is due to the existence of flat subsets of the boundary, and it willbe excluded in strongly convex domains. But to explain all this, we need a new tool: thehorospheres in general domains.

2.4.2 Horospheres

As already mentioned, the general definition of horosphere in an arbitrary bounded domainof Cn originated from the characterization of horospheres in Bn given in Proposition 2.2.20.In this section we shall introduce this new horospheres and prove their main properties;among them, new versions of Julia’s and Wolff’s lemma.

Let D be a bounded domain of Cn, and choose z0 ∈ D, x ∈ @D and R > 0. Thenthe small horosphere Ez0(x,R) and the big horosphere Fz0(x,R) of center x, pole z0 andradius R are defined by

Ez0(x,R) =©z ∈ D

ØØ lim supw→x

£kD(z, w)− kD(z0, w)

§< 1

2 log R™,

Fz0(x,R) =©z ∈ D

ØØ lim infw→x


§< 1

2 log R™.

(2.4.7)

In (2.4.7), liminf and limsup are always finite. In fact, if z0, z, w ∈ D, obviously

|kD(z, w)− kD(z0, w)| ≤ kD(z0, z);

hence for every x ∈ @D we have−1 < −kD(z0, z) ≤ lim inf

w→x


§

≤ lim supw→x


§≤ kD(z0, z) < +1.

(2.4.8)

Some remarks about (2.4.7) are in order. First of all, the pole z0 does not playany relevant role: we need it in the definition just as a normalization factor (cf. alsoLemma 2.4.11). Secondly, our definition is not linked to Cn; D could also be a hyperbolicmanifold with boundary, and nothing would change. Thirdly, the natural setting for thesedefinitions is within complete hyperbolic domains (cf. for instance Theorem 2.4.16 and thenotes to this chapter); however, they make sense in general.

(2.4.8) implies several elementary properties of the horospheres, that we collect herefor easy reference:

Lemma 2.4.10: Let D be a bounded domain of Cn, z0 ∈ D and x ∈ @D. Then:

(i) for every R > 0 we have Ez0(x,R) ⊂ Fz0(x,R);(ii) for every 0 < R1 < R2 we have Ez0(x,R1) ⊂ Ez0(x,R2) and Fz0(x,R1) ⊂ Fz0(x,R2);(iii) for every R > 1 we have Bk(z0,

12 log R) ⊂ Ez0(x,R);

(iv) for every R < 1 we have Fz0(x,R) ∩Bk(z0,−12 log R) = /∞;

(v)S

R>0Ez0(x,R) =

SR>0

Fz0(x,R) = D andT

R>0Ez0(x,R) =

TR>0

Fz0(x,R) = /∞.

Other useful properties are direct consequences of the definitions:


Lemma 2.4.11: Let D be a bounded domain of Cn, z0 ∈ D and x ∈ @D. Then:(i) if ϕ ∈ Aut(D) ∩ C0(D), then for every R > 0

ϕ°Ez0(x,R)

¢= Eϕ(z0)

°ϕ(x), R

¢and ϕ

°Fz0(x,R)

¢= Fϕ(z0)

°ϕ(x), R

¢;

(ii) if z1 ∈ D, set12 log L = lim sup

w→x

£kD(z1, w)− kD(z0, w)

§.

Then for every R > 0 we have Ez1(x,R) ⊂ Ez0(x,LR) and Fz1(x,R) ⊂ Fz0(x,LR).

Proof: (i) It is enough to remark that

kD(z, w)− kD(z0, w) = kD

°ϕ(z),ϕ(w)

¢− kD

°ϕ(z0),ϕ(w)

¢,

and that w → x iff ϕ(w) → ϕ(x).(ii) Setting

kD(z, w)− kD(z0, w) =£kD(z, w)− kD(z1, w)

§+

£kD(z1, w)− kD(z0, w)

§,

kD(z, w)− kD(z1, w) =£kD(z, w)− kD(z0, w)

§+

£kD(z0, w)− kD(z1, w)

§,

and taking the lim sup as w → x in the former equation and the lim inf as w → x in thelatter equation, we get

lim supw→x


§≤ lim sup

w→x


§+ 1

2 log L,

lim infw→x


§≥ lim inf

w→x


§− 1

2 log L,

and the assertion follows, q.e.d.

In section 2.2.2, we studied the horospheres in the unit ball Bn. We saw that there thetwo families of horospheres coincide (i.e., the limit in (2.4.7) exists), and that a horospheretouches the boundary of Bn in just one point, namely the center. Now we want to studythe horospheres in another model case, the unit polydisk ∆n of Cn, where the situation isvery different.

∆n is the unit ball for the norm

|||z||| = max©|zj |

ØØ j = 1, . . . , n™;

therefore (Corollary 2.3.7)

∀z, w ∈ ∆n k∆n(z, w) = 12 log

1 + |||∞z(w)|||1− |||∞z(w)||| ,

where∞z(w) =

µw1 − z1

1− z1w1, . . . ,

wn − zn

1− znwn

∂


is an automorphism of ∆n with ∞z(z) = 0.Since ∆n is homogeneous, we may restrict ourselves to consider only horospheres with

pole at the origin. Let x ∈ @∆n; first of all

k∆n(z, w)− k∆n(0, w) = logµ

1 + |||∞z(w)|||1 + |||w|||

∂+ 1

2 logµ

1− |||w|||2

1− |||∞z(w)|||2

∂.

Since |||∞z(x)||| = |||x||| = 1, we just have to study the behavior of the second term. Now

1− |||w|||2 = minh

©1− |wh|2

™;

1− |||∞z(w)|||2 = minj

Ω1− |zj |2

|1− zjwj |2(1− |wj |2)

æ.

Therefore1− |||w|||2

1− |||∞z(w)|||2= min

hmax

j

Ω1− |wh|21− |wj |2

· |1− zjwj |21− |zj |2

æ. (2.4.9)

Using (2.4.9) we may compute explicitely the horospheres. Since lim sup and max com-mute, we have

lim supw→x

1− |||w|||2

1− |||∞z(w)|||2= max

j

(

lim supw→x

"|1− zjwj |21− |zj |2

minh

Ω1− |wh|21− |wj |2

æ#)

= maxj

(|1− zjxj |21− |zj |2

lim supw→x

minh

Ω1− |wh|21− |wj |2

æ)

.

If |xj | < 1, we have

lim supw→x

minh

©1− |wh|2

™

1− |wj |2= 0;

therefore we ought to consider only j’s with |xj | = 1. Furthermore,

minh

Ω1− |wh|21− |wj |2

æ≤ 1;

hence

lim supw→x

1− |||w|||2

1− |||∞z(w)|||2≤ max

j

Ω|1− zjxj |21− |zj |2

ØØØØ |xj | = 1æ

. (2.4.10)

We claim that (2.4.10) is an equality. To prove this, we need to exhibit a sequencew∫ → x such that (1− |||w∫ |||2)

±(1− |||∞z(w∫)|||2) converges to the right-hand side of (2.4.10).

Set w∫ = (1− 1/∫)1/2x; we have

1− |(w∫)h|2 = (1− |xh|2) + |xh|2/∫.


Hence if |xj | = 1 we get

lim∫→1

minh

Ω1− |(w∫)h|21− |(w∫)j |2

æ= 1,

and the claim follows.To get the big horospheres, we proceed in the same way. Since lim inf and min

commute, (2.4.9) yields

lim infw→x

1− |||w|||2

1− |||∞z(w)|||2= min

h

(

lim infw→x

"

(1− |wh|2)maxj

Ω|1− zjwj |2

(1− |wj |2)(1− |zj |2)

æ#)

.

If |xh| < 1, we have

lim infw→x

"

(1− |wh|2)maxj

Ω|1− zjwj |2

(1− |wj |2)(1− |zj |2)

æ#

= +1;

therefore we must again consider only h’s with |xh| = 1. Let us fix such a h; then

lim infw→x

"

(1− |wh|2)maxj

Ω|1− zjwj |2

(1− |wj |2)(1− |zj |2)

æ#

≥ lim infw→x

|1− zhwh|21− |zh|2

=|1− zhxh|21− |zh|2

;

hence

lim infw→x

1− |||w|||2

1− |||∞z(w)|||2≥ min

h

Ω|1− zhxh|21− |zh|2

ØØØØ |xh| = 1æ

. (2.4.11)

We claim that (2.4.11) is an equality too. This time we need to exhibit a sequencew∫ → x such that (1− |||w∫ |||2)

±(1− |||∞z(w∫)|||2) converges to the right-hand side of (2.4.11).

Choose h such that |xh| = 1 and the quantity |1− zhxh|2±(1− |zh|2) is minimal, and set

(w∫)j =Ω

(1− 1/∫)1/2xj , if j 6= h;(1− 1/∫2)1/2xh, if j = h.

If j 6= h, we get

1− |(w∫)h|21− |(w∫)j |2

· |1− zj(w∫)j |21− |zj |2

=1/∫2

(1− |xj |2) + |xj |2/∫· |1− zj(w∫)j |2

1− |zj |2−→ 0

as ∫ → +1; therefore

lim∫→1

maxj

Ω1− |(w∫)h|21− |(w∫)j |2

· |1− zj(w∫)j |21− |zj |2

æ=

|1− zhxh|21− |zh|2

,

and the claim follows.In conclusion, we have proved


Proposition 2.4.12: Let x ∈ @∆n and R > 0. Then

E0(x,R) =

(

z ∈ ∆n

ØØØØØ maxj

Ω|xj − zj |21− |zj |2


< R

)

;

F0(x,R) =

(

z ∈ ∆n

ØØØØØ minj

Ω|xj − zj |21− |zj |2


< R

)

.

To better understand this result, let us look at ∆2. If we write E∆(≥, R) for thehorocycle of center ≥ and radius R in ∆, Proposition 2.4.12 says that typical examples ofhorospheres in ∆2 are

E0

°(1, 1), R

¢= E∆(1, R)×E∆(1, R), E0

°(1, 0), R

¢= E∆(1, R)×∆;

F0

°(1, 1), R

¢=

°E∆(1, R)×∆

¢∪

°∆×E∆(1, R)

¢, F0

°(1, 0), R

¢= E∆(1, R)×∆.

Figure 2.2 The horospheres in ∆2.

Therefore in the polydisk small and big horospheres are really different, and theycan touch the boundary in more than one point (see Figure 2.2). The exact situation isdescribed by:

Corollary 2.4.13: Let x ∈ @∆n and R > 0. Then:(i) E0(x,R) = F0(x,R) iff x has only one component of modulus 1;(ii) E0(x,R)∩@∆n = {x} iff x is on the Silov boundary (@∆)n of ∆n, while F0(x,R)∩@∆n

always contains {x} properly.

One may wonder if the properties of the horospheres described in Corollary 2.4.13are due to the reducibility of ∆n. This is not the case: the same situation presentsitself in bounded symmetric domains. Indeed, let D be a bounded symmetric domain,and x ∈ @D. There exists a polydisk ∆r ⊂ D (where r is the rank of the domain) suchthat x ∈ @∆r ⊂ @D and k∆r = kD|∆r×∆r (see Wolf [1972] and Abate [1987]). Hence

ED0 (x,R) ∩∆r ⊂ E∆

0 (x,R) and FD0 (x,R) ∩∆r ⊃ F∆

0 (x,R),


where E∆, ED (F∆, FD) are the small (big) horospheres of ∆r, respectively D. Therefore,by Corollary 2.4.13.(i), ED

0 (x,R) and FD0 (x,R) are, in general, different and, by Corol-

lary 2.4.13.(ii), FD0 (x,R) always touches the boundary in a set strictly bigger than {x}.

The latter phenomenon is somehow originated by the flatness of the boundary of thepolydisk. Our next aim is to prove that in strongly pseudoconvex domains this does notoccur:

Theorem 2.4.14: Let D ⊂⊂ Cn be a strongly pseudoconvex domain. Then for ev-ery z0 ∈ D, x ∈ @D and R > 0

Ez0(x,R) ∩ @D = Fz0(x,R) ∩ @D = {x}.

Proof: We begin proving that x belongs to the closure of Ez0(x,R). Let ε > 0 be given byTheorem 2.3.56; then, recalling Theorem 2.3.52, for every z, w ∈ B(x, ε) we have

kD(z, w)− kD(z0, w) ≤ 12 log

µ1 +

kz − wkd(z, @D)

∂+ 1

2 log£d(w, @D) + kz − wk

§+ K,

for a suitable constant K ∈ R depending only on x and z0. In particular, if kz − xk < εwe get

lim supw→x

[kD(z, w)− kD(z0, w)] ≤ 12 log

µ1 +

kz − xkd(z, @D)

∂+ 1

2 log kz − xk+ K. (2.4.12)

Hence if we take a sequence {z∫} ⊂ D converging to x so that {kz∫ − xk/d(z∫ , @D)} isbounded (for instance, a sequence converging non-tangentially to x), then for every R > 0we have z∫ ∈ Ez0(x,R) eventually, and thus x ∈ Ez0(x,R).

To conclude the proof, we have to show that x is the only boundary point belongingto the closure of Fz0(x,R). Suppose, by contradiction, that there exists y ∈ @D∩Fz0(x,R)with y 6= x; then we can find a sequence {zµ} ⊂ Fz0(x,R) with zµ → y.

Corollary 2.3.55 provides us with ε > 0 and K ∈ R associated to the pair (x, y); wemay assume kzµ − yk < ε for all µ ∈ N. Since zµ ∈ Fz0(x,R), we have

∀µ ∈ N lim infw→x

£kD(zµ, w)− kD(z0, w)

§< 1

2 log R;

therefore for each µ ∈ N we can find a sequence {wµ∫} ⊂ D such that lim∫→1

wµ∫ = x and

lim∫→1

£kD(zµ, wµ∫)− kD(z0, wµ∫)

§< 1

2 log R.

Moreover, we can assume kwµ∫ − xk < ε and kD(zµ, wµ∫) − kD(z0, wµ∫) < 12 log R for

all µ, ∫ ∈ N.By Corollary 2.3.55, for all µ, ∫ ∈ N we have

12 log R > kD(zµ, wµ∫)− kD(z0, wµ∫)

≥ −12 log d(zµ, @D)− 1

2 log d(wµ∫ , @D)− kD(z0, wµ∫)−K.


On the other hand, the upper boundary estimate (Theorem 2.3.51) yields c1 > 0 (inde-pendent of wµ∫) such that

∀µ, ∫ ∈ N kD(z0, wµ∫) ≤ c1 − 12 log d(wµ∫ , @D).

Therefore

∀µ ∈ N 12 log R > −1

2 log d(zµ, @D)−K − c1,

and, letting µ go to infinity, we get a contradiction, q.e.d.

As we shall see in the next section, this theorem is one of the main reasons behind thedifferent behavior of iterates in strongly convex domains with respect to weakly convexdomains.

In the first part of this book, we saw that in the classical theory of horocycles a premi-nent position is occupied by Julia’s lemma. For our horospheres, this is absolutely natural:exactly as the Kobayashi distance has a built-in Schwarz lemma, so our horospheres havea built-in Julia lemma.

To introduce this new version, look at Theorem 2.2.21, Julia’s lemma in Bn. Thattheorem can be applied to maps f ∈ Hol(Bn, Bn) such that d

°f(z), @Bn

¢/d(z, @Bn) has

finite lim inf as z → x ∈ @Bn. Now, in section 2.3.5 we have learned that in a stronglypseudoconvex domain D with base point z0 the quantity −1

2 log d(z, @D) is essentially thesame as kD(z0, z); so the hypothesis in a Julia lemma can be something like

lim infw→x

£kD(z0, w)− kD

°z0, f(w)

¢§< +1. (2.4.13)

Note that since

kD(z0, w)− kD

°z0, f(w)

¢≥ kD

°f(z0), f(w)

¢− kD

°z0, f(w)

¢≥ −kD

°z0, f(z0)

¢,

the lim inf in (2.4.13) is never −1. Moreover, if that lim inf is finite for one point z0 ∈ D,then it remains finite replacing z0 by any other point of D.

Then our most general version of Julia’s lemma is:

Proposition 2.4.15: Let D be a bounded domain of Cn, and fix a point z0 ∈ D. Letf :D → D be a holomorphic map such that there is a sequence {w∫} ⊂ D convergingto x ∈ @D so that {f(w∫)} converges to a point y ∈ @D and

lim∫→1

£kD(z0, w∫)− kD

°z0, f(w∫)

¢§≤ 1

2 log α < +1,

for a suitable α ∈ R+. Then

∀R > 0 f°Ez0(x,R)

¢⊂ Fz0(y,αR).


Proof: Let z ∈ Ez0(x,R); then

lim infw→y

£kD(f(z), w)− kD(z0, w)

§

≤ lim inf∫→1

£kD

°f(z), f(w∫)

¢− kD

°z0, f(w∫)

¢§

≤ lim inf∫→1

£kD(z, w∫)− kD

°z0, f(w∫)

¢§

≤ lim inf∫→1

£kD(z, w∫)− kD(z0, w∫)

§+ lim

∫→1


°z0, f(w∫)

¢§

≤ lim supw→x


§+ 1

2 log α

< 12 log(αR),

q.e.d.

However, as already anticipated, the horospheres are really meaningful only in com-plete hyperbolic domains (if x ∈ @D is at finite Kobayashi distance from D, then thehorospheres centered at x are just Kobayashi balls of center x). Accordingly, in completehyperbolic domains we have a neater statement:

Theorem 2.4.16: Let D ⊂⊂ Cn be complete hyperbolic. Let f ∈ Hol(D,D) and x ∈ @Dbe such that there are z0 ∈ D and α ∈ R+ so that

lim infw→x

£kD(z0, w)− kD

°z0, f(w)

¢§≤ 1

2 log α.

Then there exists y ∈ @D such that

∀R > 0 f°Ez0(x,R)

¢⊂ Fz0(y,αR). (2.4.14)

Furthermore, if D is strongly pseudoconvex then y is unique and f has non-tangentiallimit y at x.

Proof: Choose a sequence {w∫} ⊂ D converging to x such that

lim∫→1


°z0, f(w∫)

¢§= lim inf

w→x

£kD(z0, w)− kD

°z0, f(w)

¢§;

up to a subsequence, we can also assume f(w∫) → y ∈ D. Since D is complete hyperbolic,kD(z0, w∫) → +1 as ∫ → +1; therefore kD

°z0, f(w∫)

¢→ +1 as well, y must belong

to @D and we can apply Proposition 2.4.15.Finally, assume D strongly pseudoconvex, and choose y ∈ @D such that (2.4.14) holds;

it suffices to show that y is the non-tangential limit of f at x.Let {z∫} ⊂ D be a sequence converging non-tangentially to x. Then (2.4.12) implies

that for every R > 0 we have z∫ ∈ Ez0(x,R) eventually, and thus f(z∫) ∈ Fz0(y,αR) even-tually. In particular, Lemma 2.4.10.(iv) yields f(z∫) /∈ Bk(z0,−1

2 log R) eventually, and soevery limit point of {f(z∫)} belongs to the boundary of D (for D is complete hyperbolic or,if you prefer, because of Theorem 2.3.52). But then we can quote Theorem 2.4.14, statingthat the unique point belonging both to @D and to the closure of Fz0(y,αR) is y itself,concluding that f(z∫) → y and, by the arbitrariness of {z∫}, the assertion, q.e.d.


If D is not strongly pseudoconvex, the point y may not be uniquely determined,essentially because two distinct points on @D can have the same horospheres; cf. Proposi-tion 2.4.12.

As customary, Julia’s lemma will be used to study angular derivatives, as we shall seein chapter 2.7, after the introduction of new technical tools to be defined in chapter 2.6;it is now time to talk about Wolff’s lemma.

The standard Wolff lemma in Bn (and ∆) was applied to a map without fixed points.Our first Wolff’s lemma requires a bit more: we need that the sequence of iterates ofthe map under consideration be compactly divergent. In Bn this is equivalent to havingno fixed points (Proposition 2.2.30), but in general this is not enough, as shown by themap (2.4.6). Anyway, we can prove the following

Proposition 2.4.17: Let D ⊂⊂ Cn be a complete hyperbolic domain with simple bound-ary, z0 ∈ D and f ∈ Hol(D,D) such that {fk} is compactly divergent. Then there existsx ∈ @D such that

∀R > 0 f°Ez0(x,R)

¢⊂ Fz0(x,R).

In particular, if D is strongly pseudoconvex then f has non-tangential limit x at x.

Proof: Since {fk} is compactly divergent and D is complete hyperbolic,

limk→1

kD

°z0, f

k(z0)¢

= +1.

Now, we claim there is a subsequence {fk∫} such that

∀∫ ∈ N kD

°z0, f

k∫ (z0)¢

< kD

°z0, f

k∫+1(z0)¢. (2.4.15)

Indeed, let k∫ denote the largest integer k satisfying kD

°z0, fk(z0)

¢≤ ∫; then

kD

°z0, f

k∫ (z0)¢≤ ∫ < kD

°z0, f

k∫+1(z0)¢.

Up to a subsequence, we can assume that fk∫ → x ∈ @D (for {fk∫} is compactly divergentand D is bounded with simple boundary). In particular, if we set w∫ = fk∫ (z0), we havew∫ → x, f(w∫) = fk∫

°f(z0)

¢→ x and

lim inf∫→1


°z0, f(w∫)

¢§≤ 0,

by (2.4.15). Then we can apply Proposition 2.4.15 with α = 1 and y = x, and the firstassertion follows. The last assertion is an immediate consequence of Theorem 2.4.16, q.e.d.

This proposition has one main disadvantage: since, in general, Fz0(x,R) is strictly big-ger than Ez0(x,R), Proposition 2.4.17 gives no information about fk

°Ez0(x,R)

¢for k ≥ 2,

and so it is not the right statement for iteration theory.


We can obviate this disadvantage (even reintroducing the hypothesis of no fixed points)replacing the condition of simple boundary by a (strong) internal condition that we shallnow describe. Let D ⊂⊂ Cn be a bounded domain; we shall say that D is compactly ap-proximable if there exists a sequence {g∫} of holomorphic maps of D into itself convergingto the identity of D such that g∫(D) ⊂⊂ D for every ∫ ∈ N. For instance, every convexdomain D is compactly approximable: it suffices to choose z0 ∈ D, a sequence {r∫} ⊂ (0, 1)converging to 1 and set g∫(z) = z0 + r∫(z− z0). More generally, every bounded domain Dstar-shaped with respect to a point z0 ∈ D and such that z0 +r(x−z0) ∈ D for all x ∈ @Dand r ∈ (0, 1) close enough to 1 is compactly approximable.

A compactly approximable domain is topologically fairly trivial:

Lemma 2.4.18: Let D ⊂⊂ Cn be a compactly approximable domain. Then all thehomology groups Hk(D,Z) and the homotopy groups πk(D) vanish.

Proof: Let ∞ ∈ Hk(D,Z); we can find a representative of ∞ compactly supported in D.Therefore there is ∫ large enough such that (g∫)∗∞ = ∞. Now, by Corollary 2.1.32, thesequence {(g∫)k} converges to a point z0 ∈ D. But then, for k large enough, ∞ = (g∫)k

∗∞ iscontained in a contractible neighbourhood of z0, and so ∞ = 0. The same argument appliesto the homotopy groups, q.e.d.

It is then natural to conjecture that every strongly pseudoconvex topologically con-tractible domain is compactly approximable; unexpectedly, this is not true. In fact, Linand Zaıdenberg [1979] have constructed a bounded strongly pseudoconvex domain withanalytic boundary homeomorphic to the ball not compactly approximable.

Anyway, convex domains provide a sufficiently large supply of compactly approximabledomains to justify the importance of the following version of Wolff’s lemma:

Theorem 2.4.19: Let D ⊂⊂ Cn be a compactly approximable domain, and take a mapf ∈ Hol(D,D) without fixed points. Then there exists x ∈ @D such that

fk°Ez0(x,R)

¢⊂ Fz0(x,R)

for every z0 ∈ D, R > 0 and k ∈ N.

Proof: Let {g∫} be a sequence of maps converging to idD such that g∫(D) ⊂⊂ D forevery ∫ ∈ N. Set f∫ = g∫ ◦ f . By Corollary 2.1.32, every f∫ has a fixed point w∫ ∈ D. Upto a subsequence, we may assume that {w∫} converges to a point x ∈ D. If x ∈ D, then

f(x) = lim∫→1

f∫(w∫) = lim∫→1

w∫ = x,

impossible; therefore x ∈ @D.Now, for every z0, z ∈ D we have

lim∫→1

£kD(z, w∫)− kD(z0, w∫)

§≤ lim sup

w→x


§. (2.4.16)

2.4.3 Convex domains 225

Fix R > 0 and z ∈ Ez0(x,R); by (2.4.16), there are ∫0 ∈ N and ε > 0 such that

∀∫ ≥ ∫0 kD(z, w∫) < kD(z0, w∫) + 12 log R− ε.

Since w∫ is a fixed point of (f∫)k for every k ∈ N, we get

∀∫ ≥ ∫0 kD

°(f∫)k(z), w∫

¢< kD(z0, w∫) + 1

2 log R− ε.

Now, ØØkD

°(f∫)k(z), w∫

¢− kD

°fk(z), w∫

¢ØØ ≤ kD

°(f∫)k(z), fk(z)

¢−→ 0

as ∫ → +1; hence there exists ∫1 ≥ ∫0 such that

∀∫ ≥ ∫1 kD(fk(z), w∫) < kD(z0, w∫) + 12 log R− ε/2.

Therefore

lim infw→x

£kD(fk(z), w)− kD(z0, w)

§≤ lim inf

∫→1

£kD(fk(z), w∫)− kD(z0, w∫)

§< 1

2 log R,

and fk(z) ∈ Fz0(x,R), q.e.d.

In general, the point x ∈ @D, whose existence is asserted in Theorem 2.4.19, is notuniquely determined (for instance in the polydisk), and so we shall not speak of a Wolffpoint associated to the map f . However, we shall later see that x is unique if D is astrongly convex domain.

And now we can proceed to iteration theory in convex domains.

2.4.3 Convex domains

This section is devoted to the investigation of the asymptotic behavior of sequences ofiterates in bounded convex domains.

In section 2.4.1 we saw that if X is a generic taut manifold, then it is possible to findmaps f ∈ Hol(X,X) without fixed points and such that {fk} is not compactly divergent.The existence of this kind of maps is a somehow annoying phenomenon: Corollary 2.4.4is a poor replacement of Corollary 2.4.5, for one would like to study Γ(f) without relyingtoo much on the limit manifold of f .

In the ball, these maps do not exist: the sequence {fk} is compactly divergent ifff has no fixed points (Proposition 2.2.30). The next theorem shows that this happens inconvex domains too:

Theorem 2.4.20: Let D ⊂⊂ Cn be a bounded convex domain, and f ∈ Hol(D,D). Then{fk} is compactly divergent iff f has no fixed points in D.

Proof: One direction is obvious; conversely, assume that {fk} is not compactly divergent,and let ρ:D → M be the limit retraction. First of all, note that kM = kD|M×M . In fact

∀z1, z2 ∈ M kD(z1, z2) ≤ kM (z1, z2) = kM

°ρ(z1), ρ(z2)

¢≤ kD(z1, z2).


In particular, a Kobayashi ball in M is nothing but the intersection of a Kobayashi ballof D with M .

Let ϕ = f |M , and denote by Γ the closed subgroup of Aut(M) generated by ϕ; weknow, by Corollary 2.4.4, that Γ is compact. Take z0 ∈ M ; then the orbit

Γ(z0) =©∞(z0)

ØØ ∞ ∈ Γ™

is compact and contained in M . Let

C =nBk(w, r)

ØØØ w ∈ M, r > 0 and Bk(w, r) ⊃ Γ(z0)o,

where Bk(w, r) is the Kobayashi ball in D. Every Bk(w, r) is compact and convex (byProposition 2.3.46); therefore, C =

TC is a not empty compact convex subset of D. We

claim that f(C) ⊂ C.Let z ∈ C; we have to show that f(z) ∈ Bk(w, r) for every w ∈ M and r > 0 such

that Bk(w, r) ⊃ Γ(z0). Now, Bk(ϕ−1(w), r) ∈ C: in fact

Bk(ϕ−1(w), r) ∩M = ϕ−1°Bk(w, r) ∩M

¢⊃ ϕ−1

°Γ(z0)

¢= Γ(z0).

Therefore z ∈ Bk(ϕ−1(w), r) and

kD

°w, f(z)

¢= kD

≥f°ϕ−1(w)

¢, f(z)

¥≤ kD

°ϕ−1(w), z

¢≤ r,

that is f(z) ∈ Bk(w, r), as we want.In conclusion, f(C) ⊂ C; by Brouwer’s theorem, f must have a fixed point in C,

q.e.d.

This theorem, besides its importance in iteration theory, is a good tool for the con-struction of fixed points, as indicated by

Corollary 2.4.21: Let D ⊂⊂ Cn be a convex domain, and take f ∈ Hol(D,D). Let z0 bean arbitrary point of D; then f has a fixed point iff the sequence {fk(z0)} has a limit pointin D.

Proof: If {fk(z0)} has a limit point in D, the sequence {fk} cannot be compactly divergent;hence, by Theorem 2.4.20, f must have a fixed point.

Conversely, assume that f has a fixed point w ∈ D. Then the sequence {fk(z0)} iscontained in the closed Kobayashi ball of center w and radius kD(z0, w), which is compact.Hence {fk(z0)} has a limit point in D, q.e.d.

In particular, we can even generalize an argument we used in the proof of Theo-rem 2.4.20 itself:

2.4.3 Convex domains 227

Corollary 2.4.22: Let D ⊂⊂ Cn be a convex domain, and take f ∈ Hol(D,D). Assumethat there is a compact subset K of D such that f(K) ⊂ K. Then f has a fixed pointin D.

Proof: Apply Corollary 2.4.21 to a point z0 ∈ K, q.e.d.

Now we have sown enough to harvest; in fact, we are ready to prove the announcedgeneralization of the Wolff-Denjoy theorem:

Theorem 2.4.23: Let D ⊂⊂ Cn be a strongly convex domain and f ∈ Hol(D,D) withoutfixed points. Then the sequence {fk} of iterates of f converges to a point x0 ∈ @D.

Proof: Since f has no fixed points, Theorem 2.4.19 provides us with a point x ∈ @D; weclaim that fk → x. Let h ∈ Hol(D,Cn) be a limit point of {fk}; if we prove that h ≡ x,we are done.

Choose a subsequence {fk∫} converging to h. By Theorem 2.4.20, h(D) ⊂ @D; inparticular, h is constant, for D is strongly convex. By Theorem 2.4.19, for every z0 ∈ Dand R > 0 we have

∀∫ ∈ N fk∫°Ez0(x,R)

¢⊂ Fz0(x,R).

Taking the limit for ∫ → +1 we get

h°Ez0(x,R)

¢⊂ Fz0(x,R) ∩ @D = {x}

(where we are using Theorem 2.4.14), and h ≡ x, q.e.d.

Therefore Theorem 2.4.23 together with Theorems 2.4.20, 2.4.1 and Corollary 2.4.5gives a neat description of the asymptotic behavior of a sequence of iterates in a stronglyconvex domain. Note that, in particular, the point provided by Theorem 2.4.19 in stronglyconvex domains is unique, for it is the limit of the sequence of {fk}.

A careful examination of the proof of Theorem 2.4.23 indicates that we needed thestrong convexity only to quote Theorem 2.4.14, showing that the horospheres touch theboundary just in one point. Therefore, if we can somehow generalize Theorem 2.4.14 tothe weakly convex case, we can hope in a sort of Wolff-Denjoy theorem for weakly convexdomains.

Let D ⊂⊂ Cn be a convex C2 domain. For any x ∈ @D, let nx be the outer unitnormal vector to @D at x, and define ™: @D ×Cn → C by

™(x, z) = exp£(z − x,nx)

§. (2.4.17)

™ is the P -function of D; it is clear that for any x ∈ @D the function ™x = ™(x, ·) is aweak peak function for D at x.

The idea is that we can use the P -function to estimate the boundary behavior of theKobayashi distance in D, exactly as we did in section 2.3.5. To be precise, take x ∈ @D.Then the flat component F (x) of @D at x is the set

F (x) =©y ∈ @D

ØØ |™x(y)| = 1™.


In other words, the flat component at x is the intersection of @D with the real hyperplanetangent to @D at x.

A subset A of @D is saturated if F (x) ⊂ A whenever x ∈ A. The flat componentsform a closed partition of @D; since @D is compact, it is easy to see that the quotient spacewith respect to the induced equivalence relation is normal. In particular, two disjoint flatcomponents always have disjoint saturated neighbourhoods, and saturated neighbourhoodsform a fundamental system of neighbourhoods of any flat component.

Then we can prove:

Proposition 2.4.24: Let D ⊂⊂ Cn be a bounded convex C2 domain, and take x0 ∈ @Dand δ > 0. Then there exist ε0, ε1 ∈ (0, δ) with ε0 < ε1 such that there is a constant c ∈ Rso that for all z ∈ D ∩B

°F (x0), ε0

¢we have

kD

°z,D \ B

°F (x0), 2ε1

¢¢≥ −1

2 log d(z, @D) + c.

Proof: For any ε > 0 set Uε =S

x∈@D P (x, ε). Fix ε1 ∈ (0, δ) so that U2ε1 is contained in atubular neighbourhood of @D, and choose a saturated neighbourhood U ⊂⊂ B

°F (x0), ε1

¢

of F (x0). Put

Vε1 =©(x, z0) ∈ @D ×D

ØØ x ∈ U, d°z0, F (x0)

¢≥ ε1

™;

note that (x, z0) ∈ Vε1 implies z0 /∈ U .Let ™: @D ×Cn → C be the P -function of D. Since U is a saturated neighbourhood

and Vε1 is compact, there is η < 1 such that |™(x, z0)| < η < 1 for all (x, z0) ∈ Vε1 .Define φ:Vε1 ×∆ → C by

φ(x, z0, ≥) =1−™(x, z0)1−™(x, z0)

≥ −™(x, z0)1−™(x, z0)≥

,

and fix ∞ ∈ (η, 1). If we take a neighbourhood D0 ⊂⊂ Cn of D such that |™(x, z)| < ∞/ηfor all (x, z) ∈ @D×D0, then the map Φ(x, z0, z) = φ

°x, z0,™(x, z)

¢is defined and bounded

on Vε1×D0. Now choose ε0 ∈ (0, ε1) such that U2ε0 ⊂⊂ D0 and B°F (x0), 2ε0

¢⊂ U . Then

we can proceed exactly as in the proof of Theorem 2.3.54, and the assertion follows, q.e.d.

In particular we have

Corollary 2.4.25: Let D ⊂⊂ Cn be a convex C2 domain; choose two points x1, x2 ∈ @Dsuch that x1 /∈ F (x2) — and hence x2 /∈ F (x1). Then there exist ε0 > 0 and K ∈ R suchthat for any z1 ∈ D ∩B

°F (x1), ε0

¢and z2 ∈ D ∩B

°F (x2), ε0

¢we have

kD(z1, z2) ≥ −12 log d(z1, @D)− 1

2 log d(z2, @D) + K.

Proof: Mime the proof of Corollary 2.3.55, q.e.d.

Our next aim is then clear:

Notes 229

Proposition 2.4.26: Let D ⊂⊂ Cn be a convex C2 domain, and x ∈ @D. Then forevery z0 ∈ D and R > 0 we have

Fz0(x,R) ∩ @D ⊂ F (x).

Proof: The proof is identical to the second part of the proof of Theorem 2.4.14, replacingCorollary 2.3.55 by Corollary 2.4.25, q.e.d.

Summing up, we get a Wolff-Denjoy theorem for weakly convex domains:

Theorem 2.4.27: Let D ⊂⊂ Cn be a convex C2 domain, and f ∈ Hol(D,D) withoutfixed points. Choose z0 ∈ D, and let x0 ∈ D be a limit point of the sequence {fk(z0)}.Then x0 ∈ @D and

h(D) ⊂ F (x0),

where h is any limit point of the sequence of iterates of f .

Proof: Let x ∈ @D be provided by Theorem 2.4.19. Exactly as in the proof of Theo-rem 2.4.23 we see that if h is any limit point of {fk} then

h(D) ⊂ Fz0(x,R) ∩ @D ⊂ F (x),

using Proposition 2.4.26 instead of Theorem 2.4.14. In particular, x0 ∈ F (x); henceF (x) = F (x0), and we are done, q.e.d.

So in general, we cannot infer the convergence of the sequence of iterates, but at leastwe know where the images of limit points of {fk} lie. Sometimes, this can be enough toforce the convergence of the whole sequence of iterates, as shown in the following corollary,the last generalization of the Wolff-Denjoy theorem:

Corollary 2.4.28: Let D ⊂⊂ Cn be a convex C2 domain, and f ∈ Hol(D,D). Assumethere is z0 ∈ D and a point of strong convexity x0 ∈ @D such that x0 is a limit point ofthe sequence {fk(z0)}. Then the whole sequence of iterates of f converges to x0.

Proof: Clearly, f has no fixed points. Hence, by Theorem 2.4.27, every limit point of {fk}sends D into F (x0). But, since x0 is a point of strong convexity, F (x0) = {x0} and theassertion follows, q.e.d.

Notes

As far as we know, the first paper explicitely devoted to the study of the asymptotic behav-ior of a sequence of iterates in several variables is Herve [1951]. He proved close relativesof Theorem 2.4.1 and Corollaries 2.4.4 and 2.4.5 for maps of a bounded taut domain of C2

into itself. The general versions we presented are in Abate [1988c]. Vesentini [1985] stateda version of Theorem 2.4.1 for holomorphic maps sending a bounded domain of a complexBanach space into itself and having a fixed point.

An embryonal version of Corollaries 2.4.2 and 2.4.5 for domains in C2 can be foundin Lattes [1911].


The idea of the proof of Theorem 2.4.3 comes from Ca√lka [1984], where he describedconditions securing in a metric space the boundedness of the orbit of a point under theaction of a non-expansive mapping.

Theorem 2.4.6 is due to Bochner and Montgomery [1945, 1947]; Corollary 2.4.7 is inKobayashi [1970]. Theorem 2.4.8 is due to Remmert [1956, 1957]; Theorem 2.4.9 has beenproved by Kaup [1968].

The construction of the horospheres in general domains was originally motivated byProposition 2.2.20. Actually, there is a very general definition of horospheres in locallycompact complete metric spaces which generalizes both our construction and the defini-tion of horospheres in Riemannian geometry. We recall the latter: let M be a completeRiemannian manifold of nonpositive curvature, and σ:R→ M a geodesic. The Busemannfunction h associated to σ is defined by

∀z ∈ M h(z) = limt→1

£d°z,σ(t)

¢− t

§,

where d is the Riemannian distance on M , and the limit exists for trivial reasons. Thenthe horospheres associated to σ are the sublevels of h, i.e.,

E(σ, R) = {z ∈ M | h(z) < R}. (2.4.18)

Now, t = d(σ(0),σ(t)), and so (2.4.18) is akin to our definition.But the story does not end here: as already announced, there is a more general defini-

tion. Let X be a locally compact complete metric space with distance d. We may embed Xinto C0(X) mapping x ∈ X to the function dx = d(x, ·); denote by ∂(X) ⊂ C0(X) theimage. Now identify two continuous functions on X which differ only by a constant;let X be the image of the closure of ∂(X) in C0(X) under the quotient map π, andset @X = X \ π(∂(X)). It is easy to check (using the Ascoli-Arzela theorem) that Xand @X are compact in the quotient topology, and that π ◦ ∂:X → X is a homeomorphismwith the image. @X is called the ideal boundary of X.

If h ∈ @X, then h is a continuous function on X defined up to a constant. Hence thesublevels of h are well defined: they are the horospheres at the boundary point h. Now, apreimage h0 ∈ C0(X) of h ∈ @X is the limit of functions of the kind dz∫ for some sequence{z∫} ⊂ X without limit points in X. Since every π(dz∫ ) is defined up to a constant, wecan force h0 to be zero at a fixed point z0 ∈ X. This amounts to defining the horospheresassociated to h by

E(h,R) =©z ∈ X

ØØ lim∫→1

£d(z, z∫)− d(z0, z∫)

§< R

™.

and this is really quite similar to our approach. A deeper description of this kind ofhorospheres can be found in Eberlein and O’Neill [1973] and in Ballmann, Gromov andSchroeder [1985]. In this latter book it is also proved that in a complete Riemannianmanifold of nonpositive curvature the two definitions of horospheres coincide: the idealboundary of the manifold is composed only by Busemann functions.

The general properties of horospheres, but the computations in the polydisk, as wellas Theorem 2.4.14 are taken from Abate [1988a]; we anticipate that in chapter 2.6 we

Notes 231

shall prove that in strongly convex C3 domains the big and small horopheres coincide.Fadlalla [1973a, b, c], using the Caratheodory distance, introduced something similar tothe horospheres described here, and proved a version of Theorem 2.4.14 (Fadlalla [1983]).Another approach to horospheres in bounded symmetric domains is in Bassanelli [1983].

The Julia lemmas Proposition 2.4.15 and Theorem 2.4.16 are in Abate [1988f]; theywill be fundamental in chapter 2.7. Another kind of Julia’s lemma for bounded domainsin C2 proved using the Bergmann metric is in Wachs [1940]. Theorem 2.4.19 is taken fromAbate [1988a].

Theorem 2.4.20 is proved in Abate [1988e]. It would be very interesting to know howmuch it depends on the convexity of the domain. For instance, Herve [1954] has shownthat if X is a 2-dimensional simply connected taut manifold and f ∈ Hol(X,X) is not anautomorphism, then {fk} is compactly divergent iff f has no fixed points. In fact, assume{fk} is not compactly divergent, and let M be the limit manifold of f . Since f /∈ Aut(X),the dimension of M is at most 1. If dimM = 0, then f has a fixed point (even attractive:see Corollary 2.4.2). If dimM = 1, then M is a taut simply connected Riemann surface,i.e., a disk, and the assertion follows from the Wolff-Denjoy theorem. So it seems plausiblethat some sort of generalization of Theorem 2.4.20 should hold for, say, taut manifoldshomeomorphic to a ball, but the proof of such a result is at present one of the main openproblems in iteration theory. The other one is, of course, the extension of Theorem 2.4.19to more general domains.

Another proof of Theorem 2.4.20 can be achieved using the notion of aymptotic centerintroduced by Edelstein [1972]; see Kuczumow and Stachura [1989]. A very preliminaryversion of Theorem 2.4.20 can also be found in Suzuki [1987].

Theorem 2.4.23 is the main result of Abate [1988a]; the rest of section 2.4.3 comesfrom Abate [1988e]. Finally, Herve [1954] is devoted to iteration theory in ∆2.

chapter 2.4 again iteration theory -...

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